Comparison to ‘imslice’ Experiment

A pulse sequence similar to the one shown in Fig. 4.5 is used in NMR experiments to determine the power/amplitude needed for the soft pulse B1(t) in order to achieve a

certain nutation angle. The main difference is the application of a hard 90° excitation pulse at t = −(T + to), which is necessary to create transverse magnetization. This

sequence is known as ‘imslice’ in the Bruker software ‘Topspin’. By Fourier transforming the acquired signal S(t) one obtains a 1D image of the excited slice, which is weighted by the spatially dependent excitation strength of the soft pulse. The following pulse shape is used throughout the thesis for the soft excitation:

B1(t) = A sinc  3t T  exp −10  _t 2T 2! for t ∈ [−T, T ] (4.6)

with the (normalized) sinc function being defined as sinc (x) = sin (πx)

πx . (4.7)

The pulse shape has the shape of a sinc function truncated at |x| = 3 on each side with a Gaussian envelope. The pulse shall be referred to as ‘sinc3G10’ and is shown in Fig. 4.6.

In Fig. 4.7 the resulting signal and slice profile for the ‘sinc3G10’ pulse is shown for both the simulation and the corresponding NMR experiment. Good agreement is found proving the usability of the program. Furthermore, it shall be mentioned that for both the simulation and the experiment, T was set to 1 ms. This value for the soft pulse length has been used for all experiments throughout the thesis (if applicable), if not stated otherwise.

4.2 Soft pulse simulation 69 −T 0 T time −20 0 20 40 60 80 100 pulse amplitude [%] Shaped pulse B1(t) sinc3G10 Envelope

Fig. 4.6: ‘sinc3G10’ pulse and Gaussian envelope.

T T + to T + 2to time 0 signal in tensit y

(a) _{Imslice signal S(t)}

Experiment Simulation −s/2 0 s/2 z 0 signal in tensit y (b) _{1D slice profile Ft(S(t))} Experiment Simulation

Fig. 4.7: (a) Comparison of simulated signal obtained with the sequence in Fig. 4.5 and

the signal obtained from an equivalent ‘imslice’ experiment with the ‘sinc3G10’ pulse. Note, that the simulation signal is scaled such that the signal maximum coincides with that of the experiment. (b) 1D intensity profiles for simulation and experiment obtained by Fourier transform of the recorded signal from (a). The thickness of the slice is denoted by s. Similarly to (a), the simulation profile was scaled such that the maximum equals the one of the experiment.

70 Bibliography

Bibliography

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Paul Callaghan. Principles of Nuclear Magnetic Resonance Microscopy. Oxford University Press, 1993. ISBN 9780198539971.

Mattias Edén. Computer simulations in solid-state NMR. i. spin dynamics theory. Con- cepts in Magnetic Resonance Part A, 17A(1):117–154, January 2003. ISSN 1552-5023. doi: 10.1002/cmr.a.10061. URL http://onlinelibrary.wiley.com/doi/10.1002/ cmr.a.10061/abstract.

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P. R. Locher. Computer simulation of selective excitation in n.m.r. imaging. Philosophical Transactions of the Royal Society of London. B, Biological Sciences, 289(1037):537– 542, June 1980. ISSN 0962-8436, 1471-2970. doi: 10.1098/rstb.1980.0073. URL http: //rstb.royalsocietypublishing.org/content/289/1037/537. PMID: 6106229. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes,

The Art of Scientific Computing. Cambridge University Press, 3 edition, 2007.

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5 RARE Velocimetry for Cylindrical

Couette Flow

The main objective of this chapter is a description of how to accurately measure flow in a cylindrical Couette geometry using a PGSE-RARE pulse sequence. The cylindrical Cou- ette cell is a device commonly used in rheological characterization of materials. However, it is usually bulk quantities such as the bulk shear stress that are measured. Therefore, the measurement of local fluid velocities is a way of making the sample characterization more complete and can offer insights into underlying mechanism of the material under study. In particular, the focus shall be on wormlike micellar solutions that exhibit a flow phenomenon called shear banding where the flow separates in two bands of differing fluid viscosity.

As outlined in section 3.2.1.1, the flow in the Taylor-Couette cell is of cylindrical nature with the tangential velocity component being of interest for the flow measurement. For shear banded flow, this component might not only be a function of the radius r but also of the axial position z. Thus, the goal is to capture vϕ(r, z) with the 2D velocimetry

method. In essence two main difficulties have to be mastered. The first is concerning the fact that the imaging process is bound to a Cartesian grid in contrast to the cylindrical symmetry of the flow. Consequently, the sample region that is coherently excited during the pulse sequence has to be chosen carefully in order to get a quasi equivalence of a measured Cartesian flow component and vϕ(r, z). Furthermore, attention has to be paid

to the curved motion of the fluid during the k-space acquisition. The second difficulty lies in the small extent of the fluid gap for the Couette cells used in this study. In order to get a sufficient number of data points in the direction of the velocity gradient, a high resolution is necessary, which in turn leads to a small voxel volume and a potentially low SNR in each data point. To this end, alterations to the PGSE-RARE sequence are presented that allow for high resolution measurements and more accurate description of the flow dynamics.

74 5 RARE Velocimetry for Cylindrical Couette Flow

The chapter starts with a brief description of the experimental setup. Afterwards theo- retical aspects and the implementation of the PGSE-RARE pulse sequence are discussed in detail. In particular, the simulation programs described in chapter 4 are used to further the understanding of measurement artefacts. As a result, recommendations regarding the choice of measurement parameters are given. Subsequently, these recommendations are supported by experimental results. Finally, the potential of a high resolution version of the PGSE-RARE sequence is shown.

5.1 Design of Rheo-NMR Hardware

A schematic of the experimental setup is shown in Fig. 5.1. The top of the Couette cell is coupled to a drive shaft that is in turn connected to a stepper motor sitting on top of the magnet shell. The rotational frequency of the motor can be controlled via commands within the pulse sequence and thus allows for a precise setting of the applied shear rate ˙γa at any time during the experiment. This way of imposing the motion has been used

for previous work in our lab and is described in detail in Callaghan (2007) for example. It essentially corresponds to the shear-controlled mode offered by commercial rheometers. Measuring the torque/stress exerted by the fluid onto the drive-shaft is not possible in the present setup. Stress measurements for RheoNMR experiments are difficult due to the presence of high magnetic fields, although not impossible as shown by Grabowski and Schmidt (1994). The coil used for all NMR experiments was a 25 mm standard Bruker birdcage rf coil as provided by the MICRO2.5 micro imaging system.

Two different custom build cylindrical Couette cells have been used for the studies presented in this work. The first cell (A) had a total diameter of 25 mm and both cylinders were made of polyetheretherketone (PEEK). PEEK allows for an easy manipulation of surface properties and is therefore useful for studying slip effects, for example. Radii of 10.25 mm and 11.75 mm for the rotor and the inner wall of the stator respectively resulted in a gap width of 1.5 mm. In order to create a mechanical feature in the z-direction, Teflon tape was wrapped around the inner cylinder decreasing the effective gap size locally (Fig. 5.4). The z-axis of the cell is aligned with the direction of the magnetic field.

The design of the second cell (B), which was developed in our research group by a fellow PhD student (Tim Brox), was aiming for a more robust control of thermal conditions of the sample fluid as compared to cell A. Even slight temperature changes in the order of 1◦_{C can alter the behaviour of complex fluids considerably (see e.g. Berret et al. (1997)).}

Therefore active thermal control in the magnet via an air stream provided by the Bruker kit is advantageous. As cell A fills out the whole space of the 25 mm rf coil sufficient air flow

5.1 Design of Rheo-NMR Hardware 75

Fig. 5.1: Schematic of the experimental setup. A. Strain controlled stepper motor. B.

Drive shaft connecting motor and Couette cell. C. Shell of superconducting magnet. D. Cylindrical Couette cell. E. Imaging region. F. Rf coil. G. Access for air flow.

76 5 RARE Velocimetry for Cylindrical Couette Flow

around the geometry cannot be generated. Hence, thermal energy emitted by the gradient and rf coils can heat up the sample during the course of a potentially long experiment. A schematic of Couette cell B is depicted in Fig. 5.2. The total diameter of the cell has been reduced to 20 mm providing space for air to flow along the outer wall while inserted in the magnet. Spacers at the bottom and in the middle of the cell were designed to keep to the structure centred in the rf coil and to allow air to pass through. The inner radius was chosen to be 8 mm and the gap width d = 1 mm. The decrease in gap width d as compared to cell A was necessary to ensure a small variance in stress over the gap. As the material for the outer cylinder, precision bore glass tubing (GPE Scientific Limited) was preferred over PEEK, as it allowed for optical access of the fluid while outside of the magnet. This is particularly useful for the detection of air bubbles that are a common problem when dealing with complex fluids. As it can be seen in Fig. 5.2, the cell extends well beyond the coil region at the top. The coil region itself is approximately given by the position of the spacers. This allows for the minimization of end effects within the coil region that are induced by the air-fluid interface at the top of the cell. It shall be mentioned that 25 mm coil was the largest available working with both our probe and the Couette cells. Larger coils and therefore more space for the shearing geometry are available for ultra-wide bore or certain low-field magnets. For both geometries the inner cylinder was hollow to allow for the insertion of a marker fluid undergoing rigid body rotation. Marker fluids are useful for validating the accuracy of the velocimetry measurement due to the known flow profile and can be used to accurately determine the position of the gap in the recorded flow map, as shall be discussed in section 5.2.5.